Based on the Von-Karman large deflection theory of thin plates

the second-order ordinary differential Mathieu-hill Parametric vibration equation was established for of simply supported rectangular thin plate under in-plane periodic loading using the Galerkin method. The quadratic eigenvalue method was adopted to obtain the main and secondary instability domains of the rectangular thin plate with the periodic solutions of 2T and T

then the accuracy of the method was verified by finite element method

meanwhile

the influence of nonlinear elasticity on stationary amplitude caused by the main Parametric resonance was qualitatively analyzed. The analytic results showed that strong transverse parametric resonance will occur when the frequency of excitation force is about twice as large as the natural vibration frequency of the thin plate. The quadratic eigenvalue method can be used to calculate accurately those parameters

such as the frequency and the excitation coefficient of the dynamic instability region of the rectangular thin plate. As the increasing of the amplitude

the infinite increase of vibration amplitude was controlled owing to the nonlinear factor which tows system to the direction of large vibration frequency. Furthermore

the nonlinear factor can bring the system into a complex vibration state in which the vibration amplitude increased stably or rapidly.